Effect of the fluctuations around mean field for N-body systems with long range interactions

TL;DR

This paper investigates how fluctuations around the mean field influence the dynamics of large N-body systems with long-range Coulomb or gravitational interactions, deriving a fractional kinetic equation under certain assumptions.

Contribution

It introduces a statistical dynamics approach based on first principles to derive a fractional kinetic equation for such systems, connecting force behavior to fluctuation effects.

Findings

Derived a fractional kinetic equation for velocity distribution.

Linked fractional derivative order to the $1/r^2$ force behavior.

Provided a framework for understanding fluctuations in long-range interacting systems.

Abstract

We study the effect of Chandrasekhar and Holstmark's distribution of field fluctuations on the dynamics of N-body systems interacting via Coulomb or Newton gravitational force. We develop an approach based on statistical dynamics first principles whose mathematical framework is similar to the one used by Chandrasekhar and Holstmark for their field fluctuation theory. We use the Picard iteration method to approximate the Hamiltonian dynamics in the short time limit. Neglecting correlations between particles, carrying the thermodynamic limit and assuming that the system is spatially homogeneous, we find a fractional kinetic equation for the velocity distribution. Both, the fractional derivative order and the asymptotic behavior of the solution appear to be directly connected to the $1/r^2$ behavior of the Coulombian or gravitational interaction force over short distances.